www.ann-geophys.net/25/2649/2007/ © European Geosciences Union 2007
Annales
Geophysicae
Revisiting the theory of the evolution of pick-up ion distributions:
magnetic or adiabatic cooling?
H. J. Fahr
Argelander Institut f¨ur Astronomie, Astrophysics Department, Univ. of Bonn, Auf dem Huegel 71, 53121 Bonn, Germany Received: 6 September 2007 – Revised: 29 November 2007 – Accepted: 10 December 2007 – Published: 2 January 2008
Abstract. We study the phasespace behaviour of helio-spheric pick-up ions after the time of their injection as newly created ions into the solar wind bulk flow from either charge exchange or photoionization of interplanetary neutral atoms. As interaction with the ambient MHD wave fields we allow for rapid pitch angle diffusion, but for the beginning of this paper we shall neglect the effect of quasilinear or nonlin-ear energy diffusion (Fermi-2 acceleration) induced by coun-terflowing ambient waves. In the up-to-now literature con-nected with the convection of pick-up ions by the solar wind only adiabatic cooling of these ions is considered which in the solar wind frame takes care of filling the gap between the injection energy and energies of the thermal bulk of solar wind ions. Here we reinvestigate the basics of the theory be-hind this assumption of adiabatic pick-up ion reactions and correlated predictions derived from it. We then compare it with the new assumption of a pure magnetic cooling of pick-up ions simply resulting from their being convected in an interplanetary magnetic field which decreases in magnitude with increase of solar distance. We compare the results for pick-up ion distribution functions derived along both ways and can point out essential differences of observational and diagnostic relevance. Furthermore we then include stochas-tic acceleration processes by wave-parstochas-ticle interactions. As we can show, magnetic cooling in conjunction with diffu-sive acceleration by wave-particle interaction allows for an unbroken power law with the unique power indexγ=−5 be-ginning from lowest velocities up to highest energy particles of about 100 KeV which just marginally can be in resonance with magnetoacoustic turbulences. Consequences for the re-sulting pick-up ion pressures are also analysed.
Keywords. Space plasma physics (Kinetic and MHD the-ory; Transport processes; Wave-particle interactions)
Correspondence to:H. J. Fahr (hfahr@astro.uni-bonn.de)
1 Introduction
It is common knowledge since years that suprathermal pick-up ions are produced from neutral atoms all over the inner heliosphere, while their phase-space propagation is a subject much less settled in the present literature. Especially there is an ongoing debate of how efficiently pick-up ions just af-ter the pick-up process are accelerated to higher energies due to nonlinear wave-particle interactions (see e.g. Fisk, 1976a, b; Lee, 1982; Isenberg, 1987; Bogdan et al., 1991; Chalov and Fahr, 1996; Chalov and Fahr, 1998; Fahr and Lay, 2003; Chalov et al., 2004). There is perhaps some hint given by the behaviour of the solar wind proton temperature with solar distance. Namely its observed non-adiabatic temperature be-havior proves that a specific solar wind proton heating must operate in the outer heliosphere which, as meanwhile dis-cussed in length, can only be due to energy absorption from pick-up ion generated turbulence (see Smith et al., 2001; Chashei and Fahr, 2003; Chashei et al., 2003). The ionization of interstellar H-atoms penetrating the heliosphere results in the formation of keV-energetic protons in the supersonic so-lar wind regime which may be called primary pick-up ions (PUI‘s∗). The velocity distribution of these newly produced PUI‘s∗is a toroidal function which is highly anisotropic and unstable. With the free energy of this unstable distribu-tion these PUI‘s∗drive Alfvenic turbulence which by itself selfconsistently enforces pitchangle isotropization of the ini-tial velocity distribution and energy diffusion to occur (see Chalov et al., 2004, 2006).
Due to wave-wave coupling the wave energy generated by PUI‘s∗ at the injection wavelength λi=U/ p (U = solar
wind speed; p = local ion gyrofrequency) is then
Chashei, 2002; Chashei et al., 2003). From estimations it is, however, evident that only a small fraction of about 5 percent of the PUI-generated wave energy reappears in the observed proton temperature profiles, raising the question where the major portion of the wave energy produced during the pri-mary pick-up process is going.
A thermodynamic study of the solar wind proton/PUI – twofluid temperature behaviour at larger solar distances un-der consistent wave-particle energy sharing between protons and PUI‘s was carried out (Chashei et al., 2003). This study revealed the non-adiabatic proton temperature behaviour as well as a nearly isothermal pick-up ion behaviour. To clarify more quantitatively the energy branching kinetic and spec-tral details of the relevant transfer processes had to be inves-tigated. A detailed numerical study of the PUI velocity dis-tribution and the spectral Alfv´enic wave power evolution has meanwhile been carried out (Chalov et al., 2004, 2006a, b) and consisted in the simultaneous solution of a coupled sys-tem of equations consistently describing both the isotropic velocity distribution function of PUI‘s and the spectral wave power intensity.
As one can see from this study the largest portion of the self-generated wave energy is reabsorbed by PUI‘s them-selves as a result of the cyclotron resonant interaction and leads to PUI-acceleration. This just results in the energiza-tion of pick-up protons due to the stochastic acceleraenergiza-tion pro-cess probably finally producing ubiquitous power-law PUI-tails extended to energies much higher than the PUI injection energy (i.e. about 1 KeV). These tails are seen everywhere in the solar system (see Fisk et al., 2000; Fisk and Gloeckler, 2006, 2007) and they essentially reduce the amount of the wave energy which is left for absorption by solar wind pro-tons. In the following paper we shall, however, not primarily look into details of the diffusive acceleration of PUI‘s and the terms describing this phasespace propagation process, but we shall reconsider those terms in the phasespace transport equation describing processes that are effective in the redis-tribution of PUI energies from the injection energy threshold to lower energies by adiabatic ion reactions to changing mag-netic fields in the plasma box comoving with the solar wind.
2 Kinetics of magnetic cooling
In a magnetic fieldB, variable in magnitude with solar dis-tancer, freely wind-convected ions have to conserve their magnetic momentM=mv2
⊥/2B, wherem,v⊥,Bdenote the
ion mass, the component of the ion velocityvperpendicular to the magnetic field, and the interplanetary magnetic field, respectively. The gyro-averaged Lorentz force acts on these ions by decreasingv2⊥at the decrease ofB. For an isotropic distribution function this can be interpreted as an induced convection in velocity space due to a force connected with a temporal velocity changev˙m=
q
2
3h ˙v⊥i(see Fahr and Lay,
2000).
The representative Boltzmann-Vlasow equation (BVE) in the “solar” rest frame (SF) under these conditions for the sta-tionary case is given by:
(U· ∇r)f +(d dt
vm· ∇v)f =U∂f ∂r +U
∂vm
∂r ∂f
∂v (1)
=P (r, v)
while the corresponding Boltzmann-Vlasow equation (BVE) in the ”solar wind” rest frame (WF) has the following form:
∂f˜ ∂t +
1 v2
∂ ∂v(v
2
˙
vmf )˜ = ˜P (t, v) (2)
where the second term on the left hand side describes the velocity-space divergence of the phasespace flow connected with the acceleration v˙m=dvm/dt. The coordinatet=t (r)
denotes the proper time in the co-moving reference frame (WF). Furthermore P (r, v)˜ is the local ion injection rate given byP (r, v)˜ =β(r) 1
4π v2δ(v−U )withβ(r)being the lo-cal pick-up ion production rate. In contrast, in the SF, taking H-atom velocities as negligibly small with respect to the wind velocity U, the injection term is given by P (r, v)=β(r) 1
4π v2δ(v).
Furthermorev˙mdenotes the magnetic velocity decrease of
particles with a velocityv, when they are convected out with the solar wind bulk flow at a mean velocityU to larger dis-tances where the co-convected interplanetary magnetic field B is decreased. If in fact the field decreases by (1/r), as in case of the nearly azimuthal, distant Parker field, then, with the pitchangle averagehv⊥iϑof the velocity component v⊥ for pitchangle isotropic distribution functions, i.e. with
v⊥2ϑ = 23v2, one finds the followingv−dependent
mag-netic velocity-space drift
˙
vm= −U
v
r (3)
and as well its associated radial gradient ∂vm
∂r = 1
Uv˙m= − v
r (4)
As a reminder one may note, that for the more radial field at smaller distances, falling off withr−2, one simply gets an additional factor 2 in the above relations.
is seen in observations. Interestingly enough, one can, how-ever, show that the problem, seen up from these early days, is essentially solved by additionally considering the second CGL invariant, CGL2=(PkB2/ρ3), in addition to the first
CGL-invariant,CGL1=(P⊥/Bρ). HereP⊥,kandρdenote
the components of the ion pressure and the solar wind ion density, respectively. Conservation ofCGL2namely implies
an independent cooling of the velocity degree parallel to the magnetic field, i.e. ofvk(see Siewert and Fahr, 2007).
For larger distancesr≥r0=5 AU with magnetic fields
de-creasing like(1/r)this is quantitatively seen in the following way:
CGL2=(PkB2/ρ3)=
1 2ρ
D
vk2EB2 ρ3 =
1 2
D
vk2E(B ρ)
2
(5)
= 1
2( B0
ρ0r0
)2Dvk2Er2
As we have shown in Siewert and Fahr (2007) the above re-lation is fulfilled, if
vk2r2=const=C2 (6)
is valid for all individual ions. Then this above relation sim-ply requires that
dvk dr = −
2C2
vkr3 = −
vk
r (7)
This interestingly enough demonstrates that conservation of the first and second CGL invariants requires that both ve-locity degrees of freedom, i.e.vk andv⊥, cool by the same rate at the expansion of the solar wind, thus at least at larger distances tending to keep pitchangle isotropic distribution functions.
Ions which are picked up atrvwith a velocityU will, if
nothing else happens, have adiabatically, or better say “mag-netically”, cooled down to a velocityvatr, if the following relation is fulfilled:
rv(v)=
v
Ur (8)
Thus the injection of freshly created pick-up ions at rv(v)
with an initial velocityv=Uwill be responsible for ions with velocityvatr.
Taking all these constraints together one finally finds, when reminding that the time and distance coordinates are related to eachother bydr=U dt, that the solution forf˜in the WF can be given by:
˜
f = 1
2π
rβ(Uvr) U v
−3
(9) The fact that the above distribution functionf˜actually solves the BVE given by Eq. (2) is explicitly proven in the Ap-pendix A of this paper.
The above distribution function can now be further devel-oped for larger solar distances r≥r0=5 AU in the upwind
hemisphere. At large enough solar distances the upwind H-atom density can be considered as essentially constant al-lowing to assume thatnH(Uvr)=nH(r)=nH,∞, which is an
acceptable approximation for solar distancesr≥5 AU and ve-locitiesv≥0.2U. Then one evidently finds
f = r
2π Uνex,Er
2
E(
v Ur)
−2n
H,∞v−3 (10)
=νex,Er
2
EU
2π r nH,∞v
−5
The astonishing fact that one should recognize in this func-tion above is that, under pure magnetic cooling, the result-ing PUI distribution function is a power law with the inter-esting power indexα=−5, a power index which, astonish-ingly enough, was also found by Fisk and Gloeckler (2006, 2007), however, as result for the quasi-equilibrium state established between magnetoacoustically driven ion energy diffusion and magnetoacoustic turbulence generation. This seems to open up the interesting possibility that the PUI dis-tribution could perhaps be characterized by a thorough un-broken power law with the transitive uniform power index “−5” running from regions of velocities where energy diffu-sion by wave-particle interaction operates (Fisk and Gloeck-ler, 2007) down to the region where this process stops to op-erate due to non-existence of resonant conditions with the magnetoacoustic wave fields.
3 Adiabatic cooling as a contrast
As a competing approach to the above presented derivation in the up-to-now literature a phasespace transport equation has always been applied which in its general form is borrowed from the CR-transport equation originally developed for cos-mic rays (see Parker, 1965; Gleeson and Axford, 1967). This equation, though well approved for the high energy range of cosmic rays, may, however, become questionable, if it is to be applied to low energy particles like pick-up ions. This we want to demonstrate in the following.
The argumentation for the adiabatic cooling term that ap-pears in the CR transport equation (Parker, 1965; Jokipii, 1971) and also is used in the PUI transport equation (see e.g. Isenberg, 1987; Chalov et al., 1995, 1997; Mall, 2000) usually runs as follows: If the pressure of pick-up ions does work at the volume expansion, connected with the expansion of a spherically diverging solar wind flow, then thermody-namically a loss of internal pick-up ion energy4 (i.e. en-thalpy) in the comoving frame is to be expected to occur. This then results in the following thermodynamic relation
d4 dt =nV
dχ dt = −P
dV
dt (11)
conditions where adiabatic reactions of the gas in the expand-ing flow can be expected, i.e. under subsonic expansion rates and a quasi-isentropic gas behaviour.
The latter two conditions may, however, not be fulfilled in the region of supersonic solar wind expansion with no rapid relaxation processes like collisions or wave-particle in-teractions being involved (N.B.: A piston gas with a piston expanding supersonically!). That means at larger solar dis-tances, i.e. beyond 1 AU, conditions for low energy particles are in contradiction to the assumption of the validity of the above mentioned thermodynamic relation.
This is, of course, different for high energy particles of a quasi-massless species ”i” with velocitiesvior sound
veloci-tiesci=√dPi/dρimuch greater than the solar wind velocity.
These latter particles, like galactic or anomalous cosmic rays, undergoing scattering processes at stochastically distributed, magnetic inhomogeneities, quasi-frozen into the supersoni-cally expanding solar wind, will in fact behave nearly isen-tropic and adiabatic and may approximately fullfill the above thermodynamic relation (e.g. see Toptygin, 1985). This is not so, however, for low energy subsonic ions. As both exospheric solar wind theories (see Lemaire and Scherer, 1971, Marsch and Livi, 1985) and in-situ plasma observa-tions (Marsch et al., 1981) can clearly show, solar wind ions at their expansion evidently behave adiabatic and non-isentropic in the supersonic solar wind.
Nevertheless, since the isentropy assumption and the above thermodynamic relation has often been used also for low energy ions in the literature of the past (see e.g. Vasyli-unas and Siscoe, 1976; Isenberg, 1987; Chalov and Fahr, 1995, 1998; Mall, 2000) we shall look into this approach here again and want to compare it with the approach made above for pure magnetic cooling.
For a spherically symmetric radial solar wind with con-stant bulk velocityUthe change of the comoving proper vol-ume with time is given by
dV dt =(∇ ·
U)V =2U
r V (12)
and thus with the relation further above delivers the following equation
ndχ dt = −P
1 V
dV
dt = −2P U
r (13)
One now can play a trick and replace without a good physical basis macroscopic thermodynamic by corresponding kinetic quantities, taking the enthalpy per particle and the pressure in the form
χ= γ
γ−1
*
p2 2m
+
(14) and
P =nkT ≃n2 3
*
p2 2m
+
(15)
which then leads one to the relation n1
P dχ
dt ≃ 3 2
γ γ−1
d dt
p2
p2 = −2
U
r (16)
Transcribing now this macroscopic relation to properties of individual particles, not caring for probability weights only defined by the distribution function, – in general a highly problematic and questionable procedure –, will then finally yield
3 2
γ γ−1
2pdtdp p2 = −2
U
r (17)
which evidently leads to the resulting so-called adiabatic mo-mentum change given by
(dp
dt)ad= − 2 3
γ −1 γ
U
r p (18)
This strongly simplified expression has been introduced as well into the CR transport equation as also into the pick-up ion Boltzmann equation by takingγ≫1 and understanding (dpdt)ad as an “adiabatic force” acting on the comoving
parti-cles. For PUI‘s this then leads to the following BVE U∂f
∂r − 2 3U
v r
∂f
∂v =β(r) 1
2v2δ(v−U ) (19)
It has first been shown by Vasyliunas and Siscoe (1976) that this differential equation is solved by
fpui(r, v)=
3r
8π U4β(rv,ad)·(
v U)
3/2
(20) whererv,ad denotes that specific place, conjugated tor
un-der adiabatic deceleration due to the adiabatic velocity drift, which is given by
rv,ad=r·(
v U)
3/2 (21)
The above expression then translates into the following form of the distribution function whenβ(rv,ad)is expressed with
the charge exchange ionization frequency: fpui(r, v)=
3νex,ErE2
8π rU4 nH(rv,ad)·(
v U)
−3/2
(22) which for larger distances when again setting nH(rv,ad)≃nH(r)≃nH,∞ needs then to be compared
with the expression found for purely magnetic cooling (see Eq. 10) which latter when given for the normalized velocity argument(v/U )takes the form:
fpui(r, v)=
νex,ErE2
2π rU4nH,∞(
v U)
−5
i.e. “−3/2” for the adiabatic cooling instead of “−5” for the magnetic cooling. This also implies interestingly enough that under quasi-equilibrium energy diffusion driven by mag-netoacoustic turbulences (see Fisk and Gloeckler, 2007) to-gether with adiabatic cooling a broken power-law distribu-tion would result, similar to the one presented already by Isenberg (1987).
4 Distribution and pressure of wave-accelerated pick-up ions
As we have shown before pure magnetic cooling of pick-up ions leads to a power law distribution with the power index γm=−5. However, from the theory behind this power law one can conclude that it should only be governing the PUI distribution from the injection threshold downward, i.e. it should be valid for PUI velocities ofv≤U. The question thus arises how the PUI distribution might look in regions v≥U . In this region pick-up ions have definitely been ob-served (Gloeckler et al., 1993; Geiss et al., 1994; M¨obius et al., 1996, 1998) and thus they must have been transported there by acceleration processes. The idea is that this accel-eration takes place by means of diffusive Fermi-2 accelera-tion via quasilinear interacaccelera-tion of ions with either Alfvenic or magnetoacoustic turbulences (see Isenberg, 1987; Chalov and Fahr, 1998, 2000; Toptygin, 1985). In our calculations of the ongoing paper here we shall rely on the well-based ar-gument given by Fisk and Gloeckler (2006, 2007) that pick-up ions under resonant interaction with ambient magnetoa-coustic turbulences tend to develope in the range of resonant ions a saturated, unbroken power-law distribution just with a unique spectral power index ofγs=−5.
Adopting this as a fact, well supported both by theory and observations, we now calculate quantitatively the abso-lute spectral intensity of this power-law distribution from the physical principles behind. We begin with a PUI-distribution function given in the form:
fpui(r, v)=fpui,0·(v/v0)γs (24)
which yields the PUI density in the form: npui(r)=4πfpui,0(r)·
Z v∞
v0
(w/w0)−5w2dw (25)
wherefpui,0(r) is a local normalization value, andv0 and
v∞ are lower and upper velocity resonance limits of the PUI power law. These quantities have to be fixed by phys-ical constraints. From the above expression one first ob-tains the PUI density, withx=w/w0and with the assumption
x∞=v∞/v0≫1 , by
npui(r) = 4πfpui,0(r)·v03
Z x∞
1
x−3dx (26)
≃2πfpui,0(r)·v30
and the PUI pressure as formulated by Ppui(r)=4πfpui,0(r)·
m 2v
5 0
Z x∞
1
(x)−5x4dx (27)
=4πfpui,0·
m 2v
5 0ln(x∞)
As one can see, the definition of the absolute value of the PUI pressure requires the determination of the values,fpui,0,
v0andv∞, which we aim at now.
To impede PUI‘s at some inner velocity border from com-pletely migrating to lower energies by magnetic cooling (i.e. energy loss due to conservation of the magnetic mo-ment at convection to regions with smaller magnetic fields!) it should be guaranteed that they are restored with the ade-quate rate just at this lower borderv=v0 of the power-law
distribution, where energy diffusion stops to operate due to loss of resonance conditions with the turbulence. This means that near the pick-up velocity border, the particle loss rate due to magnetic cooling must be compensated by a diffusive flow due to the established energy diffusion rate. This means that the differential particle fluxes in velocity space should just be identical, i.e.(dj )m=(dj )diff.
Hereby the differential flux in velocity space due to mag-netic ion cooling at the inner velocity resonance borderv=v0
is given by
(dj )m=4π v20fpui(v0)
δv
0
δt
m
(28)
=4π v20fpui(v0)U
δv0
δr
m
When B denotes the interplanetary magnetic field which at larger distances in the ecliptic according to the Archi-median Parker spiral configuration can be assumed to be purely azimuthal and given byB=B0(r/r0)−1, then yields,
with v2⊥/B=v2sin2ϑ/B=const and the assumption of a pitchangle isotropic functionfpuileading tov⊥2ϑ=(2/3)v2,
the following relation
δv
0
δr
m = −1
2 v0
r (29)
and thus the following expression (dj )m= −2π v20fpui(v0)U
v0
r ) (30)
To keep the inner border stationary, this adiabatic loss rate needs to be compensated by the energy-diffusion gain rate given by
(dj )diff=4π v2Dvv
∂
∂vfpui (31)
whereDvvis the energy-diffusion coefficient due to
intertion with compressive magnetosonic turbulences which ac-cording to Chalov et al. (2003) is given in the form:
Dvv=Dvv,0·
U3 rE
!
v
U rE
r
3/4
whereDvv,0is a reference value atr =rE=1 AU.
(dj )diff =4π Dvv,0·
U2 rE
! rE
r
3/4
v3 ∂
∂vfpui (33) yielding
(dj )diff = −4π Dvv,0·
U2 rE
! rE
r
3/4
v35
vfpui (34) evaluating at the lower velocity boundary to
(dj )diff,0= −20π Dvv,0·
U2 rE
! rE
r
3/4
v20fpui,0 (35)
Equating adiabatic loss rate and energy diffusion rate atv=v0
then leads to
−2π v20fpui(v0)U
v0
r = (36)
−20π Dvv,0·
U2 rE
! rE
r
3/4
v20fpui,0
and thus requires that the inner border is at
v0=5Dvv,0·U
r
rE 1/4
(37) where the reference value for the diffusion coefficient ac-cording to Chalov et al. (2003) is given by
Dvv,0=
q
δU2
m
E
U rE
9Lm
(38) where
δUm2
E and Lm denote the wavenumber-average
of the magnetosonic fluctuation power at 1 AU and the magnetosonic correlation length. Taking values as those favoured by Chalov et al. (2003), i.e. Lm=3 AU and
q
δU2
m
E/U=0.5, one then finally obtains for the lower
ve-locity boundary the mildly distance-dependent value
v0=5
q
δU2
m
E
U rE
9Lm ·
U
r
rE 1/4
≃0.2U
r
rE 1/4
(39) and therewith one also finds using Eq. (26)
fpui,0(r)=npui(r)·
1
2π v03(r) (40)
= 5
3
2π U3npui(r)·
r
rE −3/4
The local PUI density npui(r) can be derived from the
H-atom densitynH(r, θ )and the [charge exchange +
photoion-ization] rate
βpui=nH(r, θ )[νpui] =nH(r, θ )[ns(r)σexU+νi] (41)
Herens(r),νpui,σex,Us,νidenote solar wind proton density,
the total H-atom ionization frequency, the charge exchange cross section, the solar wind bulk velocity and the photoion-ization frequency. The cold-model H-atom density, which is good enough for our purposes here, is given for the upwind hemisphere by (see Fahr, 1971)
nH(r, θ )=nH,∞exp[−
νpui,0r02θ
U rsinθ] (42) and thus in view of smallθ– gradients and quasi-radial PUI flow with the accordingly simplified PUI continuity equation
1 r2
d dr(r
2U n
pui)=βpui(r, θ ) (43)
leads to the following PUI density npui(r, θ )=(
r0
r )
2n pui,0+
1 r2U
Z r
r0
νpui(r, θ )r2dr (44)
5 The upper velocity resonance border
For the quantitative calculation of the higher PUI moments, like the PUI pressure, one more quantity needs to be de-fined, namely the upper velocity boundary v∞. We shall determine this quantity by asking for the condition which must be fulfilled for energy-diffusion to be operative at all. The important restriction to energy diffusion by nonlinear in-teraction with magnetosonic compressive fluctuations is that the typical diffusion periods for ions should be much larger than the associated convection periods over coherent fluctu-ation structures with coherence lengthsLmthat are given by
τcon ≃Lm/U. This leads to the requirement (see Chalov et
al., 2003) Lm≫
vλk
3U (45)
This requirement can be taken as limiting the uppermost ve-locity and hence means that the largest particle veve-locity al-lowed from this requirement is
v∞≤ 3U Lm
λk (46)
whereλkis the mean free path for particles moving parallel to the magnetic field and given by
λk=3v 8
Z +1
−1
(1−µ2)2dµ Dµµ
(47) Here Dµµ denotes the pitchangle diffusion coefficient
(µ=cosϑ denoting the pitchangle cosine) which is repre-sented by (see Schlickeiser, 1989; or Chalov and Fahr, 1998)
Dµµ∼Dvv,0VA−2
U3 rE !
v
U rE
r
3/4
withVA being the Alfv´en velocity which due toVA∼B/√ρ
at distancesr≥r0 withB∼r−1andρ∼r−2 can be taken as
constant. This clearly shows thatλk is independent of the particle velocity and thusv∞is simply given by
v∞≤ 3U Lm λkE
rE
r
3/4
(49) and reminding that the reference value ofλk atr=rE has
been found withλkE≃0.3 AU (see Chalov and Fahr, 1999) then leads to the result
v∞≤ 3Lm
λkE
rE
r
3/4
U≃30rE r
3/4
U (50)
This states thatv∞slowly falls off with solar distance and e.g. atr=r0=5 AU is given byv∞≃9.6Us. Please note that
the above relation can only be applied in the supersonic solar wind region and does not include the region downstream of the termination shock.
6 The power-law PUI pressure
On the basis of the above derived results one then obtains the PUI pressure as given by
Ppui(r)=2π m·fpui,0v05ln(x∞) (51)
=mln(x∞) 182 U
2 r
rE 1/2
npui(r)
wherex∞for general solar wind conditions is given by
x∞= v∞ 0.2Urr
E
1/4 =
Ŵ rE
r 3/4
U 0.2Urr
E
1/4 (52)
=5Ŵ·rE
r
where we have introduced the quantityŴ=Lm/λkand hence
leads to Ppui(r)=
ln(5ŴrE
r )
52
r rE
1/2
npui(r)mU2 (53)
At a solar distance ofr=r0=5 AU whithŴ=30 one thus finds
e.g. Ppui(r0)=
ln(30) 52 (5)
1/2n
pui(r)mU2 (54)
=0.306· [npui(r)mU2]
and at larger distances like 50≤r/rE≤80 from the above ex-pressions (53) and (54) one derives a ratioαpui(r)of the
lo-cal PUI pressurePpui(r)and the accumulated kinetic energy
of freshly injected PUI´s, i.e. [npui(r)mUs2], which is only
mildly variable with distance and roughly given by αpui(r)=
Ppui(r)
[npui(r)mU2] ≃
0.306 (55)
0 20 40 60 80 100
0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50 0,55 0,60
ξ = 1
ξ = 2
ξ = 3
α pui
r [AU]
Fig. 1.Shown is the functionαpuias function of the solar distance
in the upwind direction for differerent valuesŴ=Lm/λk=20 (ξ=1),
30 (ξ=2), 40 (ξ=3).
When furtheron representing the PUI pressure by a PUI tem-perature writingPpui(r)=(3/2)npuiKTpui , this then shows
that pick-up ions at their convection towards larger distances nearly behave like an isothermal fluid with
Tpui(r)≃0.204·
m KU
2
(56) a result which was also found earlier from a different ap-proach by Fahr (2002a, b).
To illustrate the more exact variation ofαpui(r)with
so-lar distance r as it results for different solar wind condi-tions during the solar activity cycle, i.e. with a typical range 20≤Ŵ=Lm/λk≤60, we have shown the plots given in Fig. 1.
It has perhaps to be mentioned here that Ŵ essentially is variable within the solar activity cycle because of the vari-ation ofLm, while the pitchangle scattering mean free path
λkdependent on the Alfvenic turbulence level may essentially be constant, i.e. 0.3≤λk≤0.9 (see Chalov and Fahr, 1995). The coherence scale Lm is essentially given by the
typi-cal distance between two consecutive high velocity humps in the solar wind velocity and, according to Richardson et al. (2001), amounts to about 3 AU≤Lm≤9 AU.
7 Solar wind deceleration by PUI pressure
In order to compare our above result for power law PUI pres-sures with earlier work (Fahr and Fichtner, 1995; Fahr and Rucinski, 1999; Fahr and Scherer, 2004) we can represent the above result in the following form
Ppui(r)=αpui(r)ρpui(r)U2 (57)
where the functionαpui(x)forx=r/rEis defined by
αpui(x)=
ln(5Ŵ/x) 52 (x)
0 20 40 60 80 100 340
350 360 370 380 390 400 410 420 430 440 450
λ = 1 * 10-3
λ = 2 * 10-3
λ = 3 * 10-3
u [kms
-1 ]
r [AU]
Fig. 2. Shown is the solar wind velocity profile resulting for dif-fererent values of the parameter3andr=30. In all cases the as-sociated dashed curve gives a comparison to the pressure-less case (i.e.αpui=0).
The solar wind deceleration resulting from the action of both the momentum loading of the solar wind by injected PUI‘s and by the PUI pressure gradient is then given by a formula derived in the Appendix B of this paper and given by U (r)=U (r0)exp[
Z x=r/rE
x=5
dx (1+2αξ )[
ξ 25rE√x +
(59) 3
2rE
αξ x +
α3 5rE
ξ− 3 rE
(1+α
5)]
whereξ denotes the local PUI abundance which in the up-wind hemisphere for distancesr≥r0=5 AU is roughly given
by
ξ =1−exp[−3(r r0−
1)] (60)
with3=σexnH∞rE. In Fig. 2 we have shown solar wind
velocity profiles in upwind direction for differerent values of the parametersŴand3.
8 Conclusions
We have shown that pick-up ions appearing as ionized H-atoms in the inner heliosphere and freshly injected into the supersonic solar wind experience magnetic cooling, rather than adiabatic cooling, after their injection into the solar wind bulk plasma. This newly derived magnetic cooling is due to the fact that ions, embedded as subsonic ions in a supersonically expanding wind, do not react adiabatically and isentropically, but at their propagation to larger distances co-moving with the bulk of the solar wind plasma they are subject only to energy-conserving pitchangle scattering and magnetic moment conservation processes. As a consequence of that they undergo magnetically induced drifts in velocity
space, thus filling the gap between the injection velocity and the thermal energies of solar wind protons. The resulting pick-up ion distribution function is a power law with a power index ofγm=−5, instead of a power indexγad=−3/2 found
for adiabatic cooling.
Including in addition stochastic pick-up ion acceleration by nonlinear interactions with magnetoacoustic turbulences at quasi-equilibrium saturation levels (see Fisk and Gloeck-ler, 2007) this then leads to an unbroken power law for pick-up ions with a unique power index ofγ=−5 valid from the lowest to the highest velocity resonance borders, i.e. from 0.5 to 100 KeV. We fix the absolute spectral intensity of the actually resulting pick-up ion power distribution by the use of its lowest velocity moments like pick-up ion density and pressure and then can calculate the pick-up ion pressure. We can show here that the ratioαpui=Ppui/[mnpuiU2]≃0.3
at distancesr≥r0=5 AU behaves as a mildly variable
quan-tity which also characterizes the pick-up ion fluid as a quasi-isothermal fluid with a temperatureTpui≃0.204·mU2/K.
As we can show in this paper, the bulk solar wind velocity is decelerated towards larger solar distances with respect to its inner asymptotic value at 5 AU. This deceleration clearly is shown in Fig. 2 and clearly is more efficient with increas-ing interstellar H-atom densitynH,∞. Though this
deceler-ation is mainly due to momentum-loading of the solar wind by PUI‘s, there is in addition an accelerative effect exerted on to the solar wind bulk due to the action of the PUI-pressure gradient. As we can also show in Fig. 2, the PUI pressure by its gradient acts counteractive to the momentum-loading and partly compensates for the effect of momentum loading. In the theoretical approach derived for the PUI pressure in this paper it, however, turns out that the effect of this pressure gradient is not as pronounced as expected from earlier the-oretical approaches (see Fahr and Fichtner, 1995; Fahr and Rucinski, 1999) in which the ratioαpuiof PUI pressure and
PUI kinetic energy density was found as a constant with a value ofαpui=0.3.
Appendix A
PUI distribution function
In the following we will confirm by introduction of the so-lution given by Eq. (10) into the BVE given by Eq. (2) valid in the WF that Eq. (6) actually is the correct solution. Intro-ducing Eq. (10) into Eq. (2) we find:
∂ ∂t(
rβ(Uvr) U v
−3)
+ 1
v2
∂ ∂v(v
2
˙
vad
rβ(Uvr) U v
−3)
(A1)
=β(r) 1
which leads to v−31
U ∂ ∂t(rβ(
v Ur))+
1 v2
∂ ∂v(v
2(
−Uv r)
rβ(Uvr) U v
−3) (A2)
=β(r) 1
2v2δ(v−U )
and v−11
U ∂ ∂t(rβ(
v Ur))−
∂ ∂v(v
3β(v
Ur)v
−3) (A3)
=β(r)1
2δ(v−U ) This furthermore is leading to
v−11 U(Uβ(
v Ur)+r
∂β ∂rv
∂rv
∂t )− ∂ ∂v(β(
v
Ur)) (A4)
=β(r)1
2δ(v−U ) and
v−1(β(v Ur)+r
∂β ∂rv
v U)−
∂β ∂rv
r
U =β(r) 1
2δ(v−U ) (A5) which finally yields
v−1β(v Ur)+
∂β ∂rv r U − ∂β ∂rv r
U =β(r) 1
2δ(v−U ) (A6) The above relation then reduces to:
β(v
Ur)=β(r) 1
2vδ(v−U ) (A7)
and evidently can be rearranged into the following from 1=β(r)·1
2 1
β(Uvr)vδ(v−U ) (A8) This above equation can then be integrated overv
Z U
0
dv=β(r)·
Z U+ǫ
U−ǫ
1
β(Uvr)vδ(v−U )dv (A9) and then evaluates to
U=β(r)·
Z U+ǫ
U−ǫ
v
β(Uvr)δ(v−U )dv=U β(r)
β(r)! (A10) This demonstrates that the PUI distribution function given in Eq. (6) in fact fulfills the Boltzmann-Vlasow equation given by Eq. (2) and thus is the solution of the problem.
Appendix B
Equation of motion of PUI-modulated winds
In the following we derive the equation of motion for the PUI-modulated solar wind. We start out with the following
equation containing the effect of the PUI-pressure gradient and the momentum loading due to pick-up of new ions by the solar wind:
ρUdU dr = −
d dr(αρiU
2)
−(mpU )(σexnHnU ) (B1)
Carrying out the derivatives then leads to UdU
dr = − 1 ρ[ρiU
2dα
dr +αU
2dρi
dr (B2)
+2αρiU
dU
dr ] −σexnHU
2
and furtheron to 1
U dU
dr (1+2αξ )= −[ξ dα dr +α
1 ρ
dρi
dr ] − 3 rE
(B3) or
dlnU dr =
1
(1+2αξ )(−ξ dα dr −α
1 ρ dξρ dr − 3 rE ) (B4) With the explicit derivative ofαpui(r)given in Eq. (53) and
x=r/rEone finds
dα dr = d dr( 1 25ln( 5Ŵ x ) √ x) (B5) = 1
25rEx[− √
x+1
2ln( 5Ŵ
x )
√
x]
= − 1
25rE√x +
1 2rE
α x and obtains
dlnU dr =
1 (1+2αξ )(
ξ 25rE√x −
ξ 2rE
α
x (B6)
−α1 ρ dξρ dr − 3 rE ) Now with the derivative ofξ(r)given in Eq. (60)
1 ρ
dξρ dr =
dξ dr +ξ
dlnρ dr = −
3 5rE
(ξ−1)+ξ(−2
r) (B7) one furthermore finds
dlnU dr =
1 (1+2αξ )(
ξ 25rE√x −
ξ 2rE α x (B8) − α3 5rE
(1−ξ )+αξ( 2 rEx
)− 3 rE
) or finally after integration:
U (r)=U (r0)exp[
Z x=r/rE
5
dx (1+2αξ )[
ξ 25rE√x
(B9) + 3 2rE αξ x + α3 5rE
ξ − 3 rE
(1+α
5)]]
References
Bogdan, T. J., Lee, M. A., and Schneider, P.: Coupled quaislinear wave-damping and stochastic acceleration of pick-up ions in the solar wind, J. Geophys. Res., 96, 161–172, 1991.
Chalov, S. V. and Fahr, H. J.: Reflection of preaccelerated pick-up ions at the solar wind termination shock: The seed for anomalous cosmci rays, Sol. Phys., 168, 389-411, 1996.
Chalov, S. V. and Fahr, H. J.: Interplanetary pick-up ion accelera-tion, Astrophys.Space Sci., 264, 509–525, 1999.
Chalov, S. V. and Fahr, H. J.: Pick-up ion acceleration at the termi-nation shock and the post-shock pick-up ion distribution, Astron. Astrophys., 360, 381–390, 2000.
Chalov, S. V., Alexashov, D. B, and Fahr, H.-J., Reabsorption of self-generated turbulent energy by pick-up protons in the outer heliosphere, Astron. Astrophys., 416, L31–L34, 2004.
Chalov, S. V., Alexashov, D. B., and Fahr, H.-J.: Interstellar pick-up protons and solar wind heating in the outer heliosphere, Astron. Lett., 32, 206–213, 2006a.
Chalov, S. V., Alexashov, D. B., and Fahr, H.-J.: Heating of the solar wind in the outer heliosphere, Astrophys. Space Sci. Trans., 2, 19–25, 2006b.
Chashei, I. V., Fahr, H. J., and Lay, G.: Heating of the solar wind ion species by wave energy dissipation, Adv. Space Res., 32(4), 507–512, 2003.
Chashei, I. V., Fahr, H. J., and Lay, G.: Ion relaxation processes in the heliospheric interface: how perturbed are ion distribution functions?, Adv. Space Res., 35, 2078–2083, 2005.
Fahr, H. J.: The interplanetary hydrogen cone and its solar cycle variations , Astron. Astrophys., 14, 263–271, 1971.
Fahr, H. J.: Global energy transfer from pick-up ions to solar wind protons, Sol. Phys., 208, 335–344, 2002.
Fahr, H. J.: Solar wind heating by an embedded quasi-isothermal pick-up ion fluid, Ann. Geophys., 20, 1509–1518, 2002, http://www.ann-geophys.net/20/1509/2002/.
Fahr, H. J. and Shizgal, B.: Modern exospheric theories and their observational relevance, Rev. Geophys. Space Phys., 21, 75–124, 1983.
Fahr, H.J. and Fichtner, H.: The influence of pick-up ion induced wave pressures on the dynamics of the mass loaded solar wind, Solar Physics, 158, 353-363, 1995.
Fahr, H. J. and Rucinski, D.: Neutral interstellar gas atoms reduc-ing the solar wind Mach number and fractionally neutralizreduc-ing the solar wind, Astron. Astrophys., 350, 1071–1078, 1999. Fahr, H. J. and Lay, G.: Remote diagnostic of the heliospheric
termination shock using neutralized post-shock pick-up ions as messengers, Astron. Astrophys., 356, 327–334, 2000.
Fahr, H. J. and Chashei, I. V.: On the thermodynamics of MHD wave-heated solar wind protons, Astron. Astrophys., 395, 991– 1000, 2002.
Fahr, H. J. and Scherer, K.: Perturbations of the solar wind flow by radial and latitudinal pick-up ion pressure gradients, Ann. Geo-phys., 22, 2229–2238, 2004,
http://www.ann-geophys.net/22/2229/2004/.
Fisk, L. A.: The acceleration of energetic particles in the interplan-etary medium by transit time damping, J. Geophys. Res., 81, 4633–4640, 1976a.
Fisk, L. A.: On the acceleration of energetic particles in the inter-planetary medium, J. Geophys. Res., 81, 4641–4645, 1976b. Fisk, L. A., Gloeckler, G., Zurbuchen, T. H., and Schwadron, N.
A.: Ubiquitous statistical acceleration in the solar wind, in: ACE Symposium 2000, edited by: Mewald, R. A., American Institute of Physics, Proceed. ACE-Symposium, edited by: Mewald, D., et al., 221–233, 2000.
Fisk, L. A. and Gloeckler, G.: The common spectrum for acceler-ated ions in the quiet-time solar wind, Astrophys. J., 640, L79– L82, 2006.
Fisk, L. A. and Gloeckler, G.: Thermodynamic constraints on stochastic acceleration in compressional turbulence, Proc. Nat. Acad. Sci., 104(3), 5749–5754, 2007.
Geiss, J., Gloeckler, G., and Mall, U.: Origin of O+pick-up ions in the heliosphere, Astron. Astrophys., 289, 933–938, 1994. Gloeckler, G., Geiss, J., Balsiger, H., Fisk, L. A., Galvin, A. B.
Ipavich, A. B., Ogilvie, K. W., von Steiger, R., and Wilken, B.: Detection of interstellar pick-up hydrogen in the solar system, Science, 261, 70–73, 1993.
Gleeson, L. J. and Axford, W. I.: Cosmic rays in the interstellar medium, Astrophys. J., 149, L115–L118, 1967.
Griffel, D. H. and Davis, L.: The anisotropy of the solar wind, Planet. Space. Sci., 17, 1009–1013, 1969.
Isenberg, P. A.: J. Geophys. Res., 92, 1067–1076, 1987.
Jokipii, J. R.: Propagation of cosmic rays in the solar wind, Rev. Geophys. Space Phys., 9, 27–87, 1971.
Lee, M. A: Coupled hydromagnetic wave excitation and ion accel-eration upstream of the earth‘s bow shock, J.Geophys.Res., 87, 5063-5071, 1982.
Lemaire, J. and Scherer, M.: Kinetic models of the solar wind, J. Geophys. Res., 76, 7479–7490, 1971.
Mall, U.: Pick-up ions in the solar wind, in: The Outer Heliosphere: Beyond the Planets, Copernicus Society, edited by: Scherer, K., Fichtner, H. and et al., 156–163, 2000.
Marsch, E., M¨uhlh¨auser, K. H., Schwenn, R., Rosenbauer,H., Pilipp, W., and Neubauer, F. M.: Solar wind protons: Three-diemnsional velcoity distributions and derived plasma parame-ters measured between 0.3 to 1.0 AU, J. Geophys. Res., 87, 52– 59, 1982.
Marsch, E. and Livi, S.: Coulomb self-collision frequencies for non-thermal velocity distributions in the solar wind, Ann. Geo-phys., 3, 545–556, 1985,
http://www.ann-geophys.net/3/545/1985/.
Moebius, E., Rucinski, D., Isenberg, P. A., and Lee, M. A.: Deter-mination of interstellar pickup ion distributions in the solar wind with SOHO and Cluster, Ann. Geophys., 14, 492–502, 1996, http://www.ann-geophys.net/14/492/1996/.
Moebius, E., Rucinski, D., Lee, M. A., and Isenberg, P. A.: De-creases in the antisunward flux of interstellar pick-up He+ asso-ciated with the radial interplanetary magnetic fields, J. Geophys. Res., 103, 257–263, 1998.
Parker, E. N.: Planet. Space Sci., 13, 9 pp., 1965.
Siewert, M. and Fahr, H. J.: A Boltzmann-kinetical description of an MHD shock with arbitrary field inclinations, Astron. Astro-phys., accepted, 2007.
Smith, C. V., Matthaeus, W. H., Zank, G. P., Ness, N. F., Oughton S., and Richardson, J.: Heating of the low latitude solar wind by dissipation of turbulent magnetic fluctuations, J. Geophys. Res., 106, 8523–8531, 2001.
